I found one related isoperimetric inequality due to Schoenberg [1].
We say that a closed curve in $\mathbb{R}^{2n}$ is convex provided it never crosses a hyperplane more than $2n$ times.
Theorem. Let $C$ be a closed convex curve in $\mathbb{R}^{2n}$ of length $L$. Let $K$ be the convex hull of $C$. Then $$ L^{2n}\geq (2\pi n)^n n! (2n)! |K|. $$
Moreover Schoenberg determined all cases of equality.
The paper of Schoenberg has 12 citations in MathSciNet and some of these papers contain generalizations of Schoenberg's result. In particular Tilli [1] (Corollary 1.3) proved
Theorem. Assume $C\subset\mathbb{R}^n$ is a compact set. Let $K$ be the convex hull of $C$. Then $$ \mathcal{H}^1(C)^n\geq (n!)^2|K|, $$ where $\mathcal{H}^1$ stands for the Hausdorff measure.
Clearly the result applies to curves in $\mathbb{R}^n$ of finite length.
See also: Maximizing an integral w.r.t. a measure on the unit sphere.
[1] Schoenberg, I. J., An isoperimetric inequality for closed curves convex in even-dimensional Euclidean spaces . Acta Math. 91, (1954). 143-164. (MathSciNet review).
[2] Tilli, P., Isoperimetric inequalities for convex hulls and related questions. . Trans. Amer. Math. Soc. 362 (2010), 4497-4509. (MathSciNet review).